A precise measurement of the $\tau$ polarisation at LEP-1

The $\tau$ polarisation has been studied with the $\eett$ data collected by the DELPHI detector at LEP in 1993, 1994 and 1995 around the $\Z$ resonance firstly through the exclusive decay channels $\mbox{\rm e}\nu\bar{\nu}$, $\mu\nu\bar{\nu}$, $\pi\nu$, $\rho\nu$ and $\aone\nu$ and secondly with an inclusive hadronic analysis which benefits from a higher efficiency and a better systematic precision. The results have been combined with those previously published on 1990 to 1992 DELPHI data, to produce results which reflect the full LEP-1 statistics. The fit of the $\tau$ polarisation dependence on the production angle yielded the polarisation parameters $\atau=0.1359 \pm 0.0096$ and $æle=0.1382 \pm 0.0116$. From these results the ratio of the vector and axial-vector effective couplings $\barvt / \barat = 0.0683\pm0.0048$ and $\barve / \barae = 0.0694\pm0.0058$ have been derived, compatible with e$-\tau$ universality. With the assumption of lepton universality,the ratio of vector to axial-vector effective couplings for leptons $\barvl / \baral = 0.0687\pm0.0037$ was obtained, implying a value of the effective weak mixing angle $\s2thw = 0.23282\pm0.00092$.